12n
0825
(K12n
0825
)
A knot diagram
1
Linearized knot diagam
4 6 10 8 2 12 11 5 6 4 7 8
Solving Sequence
7,11 4,8
5 12 1 6 10 3 2 9
c
7
c
4
c
11
c
12
c
6
c
10
c
3
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
17
+ 6u
16
+ ··· + 2b − 2, −u
17
+ 4u
16
+ ··· + 4a − 8, u
18
− 6u
17
+ ··· + 26u − 4i
I
u
2
= h11a
3
u
3
− 4u
3
a
2
+ ··· − 5a + 29,
a
3
u
3
− 5u
3
a
2
+ a
4
+ 2a
3
u − 2a
2
u
2
+ 6u
3
a − 12a
2
u + 5u
2
a − 10u
3
− a
2
+ 15au − 8u
2
+ 10a − 25u − 14,
u
4
+ u
3
+ 3u
2
+ 2u + 1i
I
u
3
= hu
11
+ 2u
10
+ 7u
9
+ 11u
8
+ 17u
7
+ 20u
6
+ 16u
5
+ 11u
4
+ u
3
− 4u
2
+ b − 4u − 2,
− 2u
11
− 2u
10
− 13u
9
− 11u
8
− 30u
7
− 21u
6
− 26u
5
− 14u
4
+ 2u
3
+ u
2
+ a + 10u + 2,
u
12
+ u
11
+ 7u
10
+ 6u
9
+ 18u
8
+ 13u
7
+ 19u
6
+ 11u
5
+ 3u
4
+ u
3
− 6u
2
− 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 46 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
=
h−u
17
+6u
16
+· · ·+2b−2, −u
17
+4u
16
+· · ·+4a−8, u
18
−6u
17
+· · ·+26u−4i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
1
4
u
17
− u
16
+ ··· −
45
4
u + 2
1
2
u
17
− 3u
16
+ ··· −
15
2
u + 1
a
8
=
1
u
2
a
5
=
−
1
4
u
17
+ u
16
+ ··· −
27
4
u + 1
−
1
2
u
17
+ 3u
16
+ ··· +
33
2
u − 3
a
12
=
−u
u
a
1
=
−u
3
− 2u
−u
5
− u
3
+ u
a
6
=
u
2
+ 1
−u
2
a
10
=
−u
17
+
11
2
u
16
+ ··· + 24u −
7
2
1
2
u
17
− 3u
16
+ ··· −
25
2
u + 2
a
3
=
1
2
u
17
−
7
2
u
16
+ ··· −
65
2
u +
13
2
−
1
2
u
17
+ 3u
16
+ ··· +
19
2
u − 2
a
2
=
−
1
2
u
16
+ 2u
15
+ ··· − 12u +
5
2
−
1
2
u
17
+ 3u
16
+ ··· +
23
2
u − 2
a
9
=
1
2
u
17
−
5
2
u
16
+ ··· −
3
2
u +
1
2
−
1
2
u
17
+ 3u
16
+ ··· +
23
2
u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
17
+ 6u
16
− 27u
15
+ 87u
14
− 227u
13
+ 492u
12
− 898u
11
+
1405u
10
−1875u
9
+ 2132u
8
−2031u
7
+ 1572u
6
−916u
5
+ 322u
4
+ 39u
3
−138u
2
+ 90u −26
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 3u
17
+ ··· + 12u − 1
c
2
, c
5
u
18
+ 12u
17
+ ··· − 96u − 16
c
3
, c
4
, c
8
c
10
u
18
− u
17
+ ··· − u + 1
c
6
, c
7
, c
11
u
18
+ 6u
17
+ ··· − 26u − 4
c
9
u
18
− u
17
+ ··· + 11u − 1
c
12
u
18
− 6u
17
+ ··· − 584u − 712
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
+ 37y
17
+ ··· − 62y + 1
c
2
, c
5
y
18
− 4y
17
+ ··· − 640y + 256
c
3
, c
4
, c
8
c
10
y
18
− 25y
17
+ ··· − 9y + 1
c
6
, c
7
, c
11
y
18
+ 18y
17
+ ··· − 28y + 16
c
9
y
18
+ 33y
17
+ ··· − 83y + 1
c
12
y
18
+ 10y
17
+ ··· + 1367744y + 506944
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.916911 + 0.561321I
a = 1.37834 − 0.77557I
b = −0.176299 + 0.948574I
13.57260 + 1.61994I 0.247785 + 0.097983I
u = 0.916911 − 0.561321I
a = 1.37834 + 0.77557I
b = −0.176299 − 0.948574I
13.57260 − 1.61994I 0.247785 − 0.097983I
u = 0.874769 + 0.633678I
a = −1.17600 + 1.10821I
b = 0.074687 − 1.124960I
13.8091 − 7.5293I 0.29015 + 4.49288I
u = 0.874769 − 0.633678I
a = −1.17600 − 1.10821I
b = 0.074687 + 1.124960I
13.8091 + 7.5293I 0.29015 − 4.49288I
u = 0.733837
a = −0.784391
b = 0.153207
−2.09100 −1.44720
u = 0.310769 + 1.286500I
a = 0.184515 − 0.432647I
b = −0.555563 + 0.718899I
1.93032 − 3.76798I 2.83080 + 5.29800I
u = 0.310769 − 1.286500I
a = 0.184515 + 0.432647I
b = −0.555563 − 0.718899I
1.93032 + 3.76798I 2.83080 − 5.29800I
u = 0.054735 + 1.390450I
a = 0.429200 + 0.371775I
b = −0.391654 − 1.269460I
5.15709 − 2.04112I 0.57723 + 3.95110I
u = 0.054735 − 1.390450I
a = 0.429200 − 0.371775I
b = −0.391654 + 1.269460I
5.15709 + 2.04112I 0.57723 − 3.95110I
u = −0.186935 + 1.380880I
a = −0.380579 − 0.441876I
b = −0.19705 + 1.46656I
1.96657 + 2.47572I 0.511687 + 1.171098I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.186935 − 1.380880I
a = −0.380579 + 0.441876I
b = −0.19705 − 1.46656I
1.96657 − 2.47572I 0.511687 − 1.171098I
u = −0.510799
a = 0.933996
b = 0.720779
−2.51252 3.72050
u = 0.280643 + 0.301264I
a = −0.588140 − 1.071850I
b = 0.030449 + 0.391403I
−0.225777 − 0.895913I −4.68203 + 7.75983I
u = 0.280643 − 0.301264I
a = −0.588140 + 1.071850I
b = 0.030449 − 0.391403I
−0.225777 + 0.895913I −4.68203 − 7.75983I
u = 0.33904 + 1.59351I
a = −0.148834 + 1.046660I
b = 0.94820 − 2.81595I
−18.8931 − 3.0610I 2.60805 + 0.97919I
u = 0.33904 − 1.59351I
a = −0.148834 − 1.046660I
b = 0.94820 + 2.81595I
−18.8931 + 3.0610I 2.60805 − 0.97919I
u = 0.29855 + 1.60658I
a = −0.023307 − 1.135190I
b = −0.66976 + 3.18283I
−18.3049 − 11.9078I 2.47969 + 4.91668I
u = 0.29855 − 1.60658I
a = −0.023307 + 1.135190I
b = −0.66976 − 3.18283I
−18.3049 + 11.9078I 2.47969 − 4.91668I
6
II. I
u
2
= h11a
3
u
3
− 4u
3
a
2
+ · · · − 5a + 29, a
3
u
3
− 5u
3
a
2
+ · · · + 10a −
14, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
a
−0.354839a
3
u
3
+ 0.129032a
2
u
3
+ ··· + 0.161290a − 0.935484
a
8
=
1
u
2
a
5
=
−0.354839a
3
u
3
+ 0.129032a
2
u
3
+ ··· + 1.16129a − 0.935484
0.677419a
3
u
3
− 0.0645161a
2
u
3
+ ··· + 0.419355a − 0.0322581
a
12
=
−u
u
a
1
=
−u
3
− 2u
u
3
− u
2
− 1
a
6
=
u
2
+ 1
−u
2
a
10
=
−a
2
u
0.193548a
3
u
3
− 1.16129a
2
u
3
+ ··· + 0.548387a − 0.580645
a
3
=
−a
3
u
2
+ a
0.387097a
3
u
3
− 0.322581a
2
u
3
+ ··· + 0.0967742a − 1.16129
a
2
=
0.0967742a
3
u
3
− 0.580645a
2
u
3
+ ··· + 0.774194a − 1.29032
0.0967742a
3
u
3
+ 0.419355a
2
u
3
+ ··· − 0.225806a − 1.29032
a
9
=
0.548387a
3
u
3
− 0.290323a
2
u
3
+ ··· + 0.387097a + 1.35484
−0.451613a
3
u
3
− 0.290323a
2
u
3
+ ··· + 0.387097a − 2.64516
(ii) Obstruction class = −1
(iii) Cusp Shapes =
24
31
a
3
u
3
+
84
31
a
3
u
2
−
20
31
u
3
a
2
+
96
31
a
3
u −
8
31
a
2
u
2
+
16
31
u
3
a +
40
31
a
3
+
44
31
a
2
u +
56
31
u
2
a −
68
31
u
3
+
8
31
a
2
+
64
31
au −
52
31
u
2
+
68
31
a −
148
31
u −
10
31
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 7u
15
+ ··· − 478u + 73
c
2
, c
5
(u
2
− u + 1)
8
c
3
, c
4
, c
8
c
10
u
16
+ u
15
+ ··· − 24u + 79
c
6
, c
7
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)
4
c
9
u
16
+ 3u
15
+ ··· + 180u + 79
c
12
(u
4
+ u
3
+ 5u
2
− u + 2)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 27y
15
+ ··· + 40448y + 5329
c
2
, c
5
(y
2
+ y + 1)
8
c
3
, c
4
, c
8
c
10
y
16
− 21y
15
+ ··· − 51768y + 6241
c
6
, c
7
, c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
4
c
9
y
16
+ 23y
15
+ ··· + 58924y + 6241
c
12
(y
4
+ 9y
3
+ 31y
2
+ 19y + 4)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = 0.918767 + 0.732292I
b = −1.333200 − 0.320958I
4.72380 − 0.61478I 0.17326 − 1.44464I
u = −0.395123 + 0.506844I
a = 0.17223 − 1.96502I
b = 0.571429 + 0.327901I
4.72380 + 3.44499I 0.17326 − 8.37284I
u = −0.395123 + 0.506844I
a = 1.07219 + 1.87866I
b = −0.28348 − 1.48247I
4.72380 + 3.44499I 0.17326 − 8.37284I
u = −0.395123 + 0.506844I
a = −1.61576 − 1.76681I
b = 0.189337 + 0.648876I
4.72380 − 0.61478I 0.17326 − 1.44464I
u = −0.395123 − 0.506844I
a = 0.918767 − 0.732292I
b = −1.333200 + 0.320958I
4.72380 + 0.61478I 0.17326 + 1.44464I
u = −0.395123 − 0.506844I
a = 0.17223 + 1.96502I
b = 0.571429 − 0.327901I
4.72380 − 3.44499I 0.17326 + 8.37284I
u = −0.395123 − 0.506844I
a = 1.07219 − 1.87866I
b = −0.28348 + 1.48247I
4.72380 − 3.44499I 0.17326 + 8.37284I
u = −0.395123 − 0.506844I
a = −1.61576 + 1.76681I
b = 0.189337 − 0.648876I
4.72380 + 0.61478I 0.17326 + 1.44464I
u = −0.10488 + 1.55249I
a = 0.084078 − 0.977337I
b = 0.62521 + 3.74384I
11.72550 + 5.19385I 3.82674 − 6.02890I
u = −0.10488 + 1.55249I
a = −0.864979 + 0.796080I
b = 0.82603 − 1.86134I
11.72550 + 5.19385I 3.82674 − 6.02890I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.10488 + 1.55249I
a = 0.050829 + 1.247660I
b = −0.44287 − 3.34396I
11.72550 + 1.13408I 3.82674 + 0.89930I
u = −0.10488 + 1.55249I
a = 0.182649 − 0.480753I
b = 1.34755 + 1.14590I
11.72550 + 1.13408I 3.82674 + 0.89930I
u = −0.10488 − 1.55249I
a = 0.084078 + 0.977337I
b = 0.62521 − 3.74384I
11.72550 − 5.19385I 3.82674 + 6.02890I
u = −0.10488 − 1.55249I
a = −0.864979 − 0.796080I
b = 0.82603 + 1.86134I
11.72550 − 5.19385I 3.82674 + 6.02890I
u = −0.10488 − 1.55249I
a = 0.050829 − 1.247660I
b = −0.44287 + 3.34396I
11.72550 − 1.13408I 3.82674 − 0.89930I
u = −0.10488 − 1.55249I
a = 0.182649 + 0.480753I
b = 1.34755 − 1.14590I
11.72550 − 1.13408I 3.82674 − 0.89930I
11
III.
I
u
3
= hu
11
+2u
10
+· · ·+b−2, −2u
11
−2u
10
+· · ·+a+2, u
12
+u
11
+· · ·−2u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
2u
11
+ 2u
10
+ ··· − 10u − 2
−u
11
− 2u
10
+ ··· + 4u + 2
a
8
=
1
u
2
a
5
=
2u
11
+ u
10
+ 12u
9
+ 5u
8
+ 25u
7
+ 9u
6
+ 18u
5
+ 6u
4
− 5u
3
+ u
2
− 8u
−u
11
− u
10
− 6u
9
− 5u
8
− 12u
7
− 9u
6
− 8u
5
− 6u
4
+ 2u
3
+ 2u + 1
a
12
=
−u
u
a
1
=
−u
3
− 2u
−u
5
− u
3
+ u
a
6
=
u
2
+ 1
−u
2
a
10
=
−2u
11
− u
10
+ ··· + 12u + 1
u
11
+ 5u
9
+ 9u
7
+ u
6
+ 6u
5
+ 3u
4
− u
3
+ 3u
2
− 2u − 1
a
3
=
−u
11
− u
10
+ ··· + 7u − 2
−u
11
− 4u
9
+ u
8
− 3u
7
+ 4u
6
+ 5u
5
+ 4u
4
+ 5u
3
− u
2
− 3u − 1
a
2
=
−u
11
− u
10
− 7u
9
− 5u
8
− 17u
7
− 8u
6
− 15u
5
− 3u
4
+ u
3
+ u
2
+ 5u − 2
u
10
+ 2u
9
+ 6u
8
+ 9u
7
+ 12u
6
+ 12u
5
+ 7u
4
+ 2u
3
− 3u
2
− 4u − 1
a
9
=
−u
11
− 7u
9
− u
8
− 18u
7
− 4u
6
− 18u
5
− 5u
4
− 2u
2
+ 9u
−u
10
− 4u
8
+ u
7
− 4u
6
+ 3u
5
+ 2u
4
+ 2u
3
+ 4u
2
− 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
11
+ 3u
10
+ 13u
9
+ 15u
8
+ 28u
7
+ 20u
6
+ 16u
5
− 5u
4
− 15u
3
− 19u
2
− 13u + 1
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 3u
11
+ ··· − 3u − 1
c
2
u
12
+ 3u
11
+ ··· − 3u − 1
c
3
, c
8
u
12
+ u
11
+ ··· − 10u
2
+ 1
c
4
, c
10
u
12
− u
11
+ ··· − 10u
2
+ 1
c
5
u
12
− 3u
11
+ ··· + 3u − 1
c
6
, c
7
u
12
+ u
11
+ ··· − 2u − 1
c
9
u
12
+ u
11
+ 4u
10
+ 2u
9
+ 6u
8
+ u
7
+ 6u
6
− u
5
+ u
4
+ 4u
3
+ 3u
2
− 1
c
11
u
12
− u
11
+ ··· + 2u − 1
c
12
u
12
+ u
11
+ u
10
− 3u
9
− 11u
8
+ 8u
7
+ 18u
6
− 9u
5
− 12u
4
− 9u
3
− 7u
2
− 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 3y
11
+ ··· − 5y + 1
c
2
, c
5
y
12
− 5y
11
+ ··· + 3y + 1
c
3
, c
4
, c
8
c
10
y
12
− 15y
11
+ ··· − 20y + 1
c
6
, c
7
, c
11
y
12
+ 13y
11
+ ··· + 8y + 1
c
9
y
12
+ 7y
11
+ ··· − 6y + 1
c
12
y
12
+ y
11
+ ··· + 14y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.840044
a = 1.57497
b = −0.211629
0.654628 −1.77310
u = −0.042760 + 1.221860I
a = 0.288630 + 0.954256I
b = −1.50897 − 1.14119I
7.97414 − 1.59154I 3.95302 + 1.55241I
u = −0.042760 − 1.221860I
a = 0.288630 − 0.954256I
b = −1.50897 + 1.14119I
7.97414 + 1.59154I 3.95302 − 1.55241I
u = −0.375195 + 1.251700I
a = −0.224979 − 0.971377I
b = 0.70872 + 1.67932I
4.52363 + 4.37134I 1.55173 − 4.04786I
u = −0.375195 − 1.251700I
a = −0.224979 + 0.971377I
b = 0.70872 − 1.67932I
4.52363 − 4.37134I 1.55173 + 4.04786I
u = 0.274032 + 1.321170I
a = 0.223039 − 0.246206I
b = 0.192113 + 0.799962I
1.21953 − 3.32394I −5.93901 + 2.04027I
u = 0.274032 − 1.321170I
a = 0.223039 + 0.246206I
b = 0.192113 − 0.799962I
1.21953 + 3.32394I −5.93901 − 2.04027I
u = 0.646284
a = −0.471074
b = −0.501207
−2.99063 −14.9120
u = −0.08539 + 1.54129I
a = −0.324208 + 0.996147I
b = −0.47764 − 2.85858I
11.88070 + 3.38261I 4.53422 − 1.77544I
u = −0.08539 − 1.54129I
a = −0.324208 − 0.996147I
b = −0.47764 + 2.85858I
11.88070 − 3.38261I 4.53422 + 1.77544I
15
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.173805 + 0.368979I
a = −0.51443 − 3.31625I
b = 0.942190 + 0.803863I
5.17878 + 2.27587I 4.24271 − 2.34851I
u = −0.173805 − 0.368979I
a = −0.51443 + 3.31625I
b = 0.942190 − 0.803863I
5.17878 − 2.27587I 4.24271 + 2.34851I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
+ 3u
11
+ ··· − 3u − 1)(u
16
− 7u
15
+ ··· − 478u + 73)
· (u
18
+ 3u
17
+ ··· + 12u − 1)
c
2
((u
2
− u + 1)
8
)(u
12
+ 3u
11
+ ··· − 3u − 1)(u
18
+ 12u
17
+ ··· − 96u − 16)
c
3
, c
8
(u
12
+ u
11
+ ··· − 10u
2
+ 1)(u
16
+ u
15
+ ··· − 24u + 79)
· (u
18
− u
17
+ ··· − u + 1)
c
4
, c
10
(u
12
− u
11
+ ··· − 10u
2
+ 1)(u
16
+ u
15
+ ··· − 24u + 79)
· (u
18
− u
17
+ ··· − u + 1)
c
5
((u
2
− u + 1)
8
)(u
12
− 3u
11
+ ··· + 3u − 1)(u
18
+ 12u
17
+ ··· − 96u − 16)
c
6
, c
7
((u
4
− u
3
+ 3u
2
− 2u + 1)
4
)(u
12
+ u
11
+ ··· − 2u − 1)
· (u
18
+ 6u
17
+ ··· − 26u − 4)
c
9
(u
12
+ u
11
+ 4u
10
+ 2u
9
+ 6u
8
+ u
7
+ 6u
6
− u
5
+ u
4
+ 4u
3
+ 3u
2
− 1)
· (u
16
+ 3u
15
+ ··· + 180u + 79)(u
18
− u
17
+ ··· + 11u − 1)
c
11
((u
4
− u
3
+ 3u
2
− 2u + 1)
4
)(u
12
− u
11
+ ··· + 2u − 1)
· (u
18
+ 6u
17
+ ··· − 26u − 4)
c
12
(u
4
+ u
3
+ 5u
2
− u + 2)
4
· (u
12
+ u
11
+ u
10
− 3u
9
− 11u
8
+ 8u
7
+ 18u
6
− 9u
5
− 12u
4
− 9u
3
− 7u
2
− 1)
· (u
18
− 6u
17
+ ··· − 584u − 712)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
+ 3y
11
+ ··· − 5y + 1)(y
16
+ 27y
15
+ ··· + 40448y + 5329)
· (y
18
+ 37y
17
+ ··· − 62y + 1)
c
2
, c
5
((y
2
+ y + 1)
8
)(y
12
− 5y
11
+ ··· + 3y + 1)
· (y
18
− 4y
17
+ ··· − 640y + 256)
c
3
, c
4
, c
8
c
10
(y
12
− 15y
11
+ ··· − 20y + 1)(y
16
− 21y
15
+ ··· − 51768y + 6241)
· (y
18
− 25y
17
+ ··· − 9y + 1)
c
6
, c
7
, c
11
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
4
)(y
12
+ 13y
11
+ ··· + 8y + 1)
· (y
18
+ 18y
17
+ ··· − 28y + 16)
c
9
(y
12
+ 7y
11
+ ··· − 6y + 1)(y
16
+ 23y
15
+ ··· + 58924y + 6241)
· (y
18
+ 33y
17
+ ··· − 83y + 1)
c
12
((y
4
+ 9y
3
+ 31y
2
+ 19y + 4)
4
)(y
12
+ y
11
+ ··· + 14y + 1)
· (y
18
+ 10y
17
+ ··· + 1367744y + 506944)
18
